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Theorem sb4bOLD 2490
 Description: Obsolete version of sb4b 2489 as of 21-Feb-2024. (Contributed by NM, 27-May-1997.) Revise df-sb 2071. (Revised by Wolf Lammen, 25-Jul-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sb4bOLD (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑)))

Proof of Theorem sb4bOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfeqf2 2385 . . . 4 (¬ ∀𝑥 𝑥 = 𝑡 → Ⅎ𝑥 𝑦 = 𝑡)
2 nfnf1 2156 . . . . . . 7 𝑥𝑥 𝑦 = 𝑡
3 id 22 . . . . . . 7 (Ⅎ𝑥 𝑦 = 𝑡 → Ⅎ𝑥 𝑦 = 𝑡)
42, 3nfan1 2199 . . . . . 6 𝑥(Ⅎ𝑥 𝑦 = 𝑡𝑦 = 𝑡)
5 equequ2 2034 . . . . . . . 8 (𝑦 = 𝑡 → (𝑥 = 𝑦𝑥 = 𝑡))
65imbi1d 346 . . . . . . 7 (𝑦 = 𝑡 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑡𝜑)))
76adantl 486 . . . . . 6 ((Ⅎ𝑥 𝑦 = 𝑡𝑦 = 𝑡) → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑡𝜑)))
84, 7albid 2223 . . . . 5 ((Ⅎ𝑥 𝑦 = 𝑡𝑦 = 𝑡) → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑)))
98pm5.74da 804 . . . 4 (Ⅎ𝑥 𝑦 = 𝑡 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑))))
101, 9syl 17 . . 3 (¬ ∀𝑥 𝑥 = 𝑡 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑))))
1110albidv 1922 . 2 (¬ ∀𝑥 𝑥 = 𝑡 → (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑))))
12 df-sb 2071 . 2 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
13 ax6ev 1973 . . . 4 𝑦 𝑦 = 𝑡
1413a1bi 367 . . 3 (∀𝑥(𝑥 = 𝑡𝜑) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)))
15 19.23v 1944 . . 3 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)))
1614, 15bitr4i 281 . 2 (∀𝑥(𝑥 = 𝑡𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)))
1711, 12, 163bitr4g 318 1 (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1537  ∃wex 1782  Ⅎwnf 1786  [wsb 2070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-12 2176  ax-13 2380 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-ex 1783  df-nf 1787  df-sb 2071 This theorem is referenced by: (None)
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